AMC 10 · 2015 · #11
Grade 8 geometry-2dPick an answer.
The problem is about a line and the axes, so tool #1 (Draw a Diagram) comes first: plotting the intercepts pins down the three corners and reveals a right triangle resting against the axes. Once the shape is a right triangle with legs on the axes, the three altitudes split into easy pieces, so tool #7 (Identify Subproblems) handles them one at a time — the two legs are themselves altitudes, and the third altitude (to the slanted hypotenuse) comes from the area. Tool #3 (Eliminate Possibilities) is a safety net: only one answer choice has a denominator of 13, which is exactly what the hypotenuse altitude forces.
Find the three corners
The intercepts give the three corners.
A line meets an axis exactly where the other coordinate is zero, so setting one variable to 0 finds each intercept.
6.EE.B.7Draw A DiagramSee the right triangle and its hypotenuse
It is a familiar right triangle with hypotenuse 13.
Two sides lying on the perpendicular axes must form a right angle, so the Pythagorean theorem gives the slanted side.
8.G.B.7Draw A DiagramTwo altitudes are the legs themselves
Two altitudes are the legs themselves.
In a right triangle the two legs are already perpendicular, so each leg is the height when the other leg is the base.
6.G.A.1Identify SubproblemsThird altitude from the area
The area gives the third altitude.
A triangle's area is fixed, so a longer base forces a shorter matching height.
A triangle's area is fixed, so a longer base forces a shorter matching height.
▸ Why?
An area is half the base times the height, so with the area fixed the two are locked against each other.
▸ Why?
Because that trade never turns around, the longest side always carries the shortest altitude.
Add the three altitudes
Adding gives 281/13, choice (E).
Writing the whole numbers over 13 lets all three altitudes share one denominator and add directly.
5.NF.A.1Eliminate PossibilitiesIn a right triangle the two legs are already altitudes; for the slanted side, use the fact that the area stays the same to find its matching height.
- Find the three corners
- See the right triangle and its hypotenuse
- Two altitudes are the legs themselves
- Third altitude from the area
- Add the three altitudes